A simple pendulum fixed inside a car has a time period of 4 seconds while the car is travelling uniformly on a horizontal road. Upon pressing the accelerator, the time period shifts to 3.99 seconds. Determine the acceleration of the car using an approximate analysis.

Step-by-step solution

The initial time period of the simple pendulum is given by: T₁ = 2 L g = 4 When the car accelerates with acceleration a , the effective acceleration due to gravity becomes: g_ eff = g^2 + a^2 = g (1 + a^2 g^2 )^ 1/2 The new time period is given by: T₂ = 2 L g_ eff = 3.99 Taking the ratio of the time periods: T₂ T₁ = ( g g_ eff )^ 1/2 = ( 1 + a^2 g^2 )^ -1/4 Since the change in time period is very small, a is small compared to g . Using the binomial approximation (1 + x)^n 1 + nx : T₂ T₁ 1 - 1 4 a^2 g^2 Substituting